Big money managers led by Fidelity Investments are close to launching a private trading venue designed to let them buy and sell large blocks of stock without the involvement of Wall Street firms and high-speed traders, according to people familiar with the matter.

Nine firms including BlackRock Inc., Bank of New York Mellon Corp., J.P. Morgan Chase & Co. and T. Rowe Price Group Inc., are forming a company that will operate a “dark...